The redshift the Sun's boiling hides

In 1907, eight years before he completed general relativity, Einstein used the equivalence principle alone to predict that a clock low in a gravitational field runs slow compared with one higher up, and that light climbing out of the field therefore arrives with a lower frequency. The clock that runs slow lower down follows the argument; the parallelogram that will not close shows why no theory that keeps energy conservation can avoid it. For the Sun the prediction is a number: light leaving the solar surface should arrive at the Earth redshifted by 633 metres a second, expressed as the Doppler velocity that would give the same shift. Einstein proposed looking for it in the solar spectrum, which was the only strong gravitational field anyone could see.

The prediction was confirmed in the laboratory by Pound and Rebka in 1959, across 22.5 metres of tower, and by white dwarfs, whose shifts are a hundred times larger. The Sun, where it was first proposed, held out the longest. That is not because the effect is small. Solar spectroscopists in the 1910s and 1920s were measuring line positions to tens of metres a second. The difficulty was that different lines gave different answers, ranging from much less than the prediction to roughly the prediction, with a systematic pattern nobody could at first explain. The Sun’s surface turned out to be producing a second shift of the same size, and the history of the measurement is the history of learning to measure that second shift well enough to subtract it.

Six hundred and thirty-three metres a second #

The arithmetic of the gravitational part is simple, and it is worth doing exactly because every term is known far better than any measurement.

The fractional shift of light climbing from the solar surface to infinity is $GM_\odot/R_\odot c^2 = 2.12 \times 10^{-6}$, which multiplied by the speed of light is 636.3 metres a second. The light does not climb to infinity: it stops at the Earth’s orbit, which is itself a little way down the Sun’s well, so 2.96 metres a second of the climb is never made. It then falls into the Earth’s own field and gains back 0.21 metres a second. The net gravitational shift, as received by a spectrograph on the ground, is 633.1 metres a second. The Earth’s orbital motion adds a further small term, a time dilation of the observer’s own clock, which is removed with the ordinary correction for the observer’s velocity.

Against those numbers sits a second contribution, drawn below them in the budget, which is the subject of the rest of this essay: the motion of the solar surface itself, which shifts a line formed near the bottom of the visible layer by about two hundred metres a second towards the blue and one formed higher by much less. The gravitational shift is the same for every line, because every photon climbs out of the same well. The surface’s contribution is not, and the spread in the old measurements was its signature.

The order of magnitude is the whole difficulty. A nuisance a hundredth the size of the signal can be handled crudely; one comparable with it cannot be handled at all unless it is understood. And it can only be understood by modelling something that the measurement was not designed to look at, namely the structure of the solar atmosphere. That is the general shape of an experiment limited by a systematic error rather than by noise: the precision is ample, and the accuracy depends entirely on a model of something else.

A surface that boils #

The visible surface of the Sun, the photosphere, is the top of its convection zone. Below it, heat from the interior — the light that takes a hundred thousand years to leave by diffusion through the radiative zone — is carried the last two hundred thousand kilometres by convection, as in a pan of soup. At the surface the convection is seen directly, in a telescope with good seeing, as granulation: a pattern of bright cells about a thousand kilometres across, separated by narrow dark lanes, each cell lasting a few minutes. Hot gas rises in the centre of each granule at about a kilometre a second, radiates its heat away at the surface, cools, spreads sideways and sinks again in the lanes.

The rising gas is hotter, and so brighter, than the sinking gas. That asymmetry is what does the damage. A spectral line seen from the Earth is the sum of the light from millions of granules and lanes, and each part of the surface contributes in proportion to its brightness. The rising gas, moving towards the observer, contributes more light than the sinking gas moving away, so the line’s average is pulled towards the blue, even though as much mass goes down as comes up.

The model drawn above is the simplest one that has the mechanism in it: two streams, one rising and bright, one sinking and faint, each absorbing light at its own Doppler shift. The rising gas covers sixty per cent of the surface, is half as bright again as the lanes, and comes up at a kilometre a second. Because mass is conserved, the narrower lanes must carry the same flux down, at a kilometre and a half a second. The line seen is the sum of the two streams’ absorption, each weighted by its brightness.

The sum is not simply a shifted copy of a symmetric line. It is asymmetric, with its deepest point dominated by the bright rising gas and its wings carrying more of the faint sinking gas, and that asymmetry has two consequences. The first is that the shift of the line depends on which point of it is measured. The deepest point lies 615 metres a second to the blue; the centre of gravity of the absorbed light lies 232 metres a second to the blue. A spectroscopist fitting a symmetric profile to the line gets a third answer, somewhere between. The second consequence is the shape traced by the midpoints of the line at successive depths — the line bisector, a standard diagnostic since the 1980s — which leans steadily redward from the core towards the continuum. Real solar lines show this lean in their upper parts and add a hook near the core, so their bisectors are shaped like a letter C; the hook needs more than two streams to reproduce. Either way, the bisector of a line is a direct picture of the convective motions and says nothing about gravity.

The deeper lesson is about zero points. The note that changes on approach measures a Doppler shift as the displacement of a line from where it would be at rest. That presupposes a line with a well-defined position. A line made of two streams has several positions, each with a defensible definition, and they disagree by hundreds of metres a second. For a measurement aiming at 633 metres a second, the definition of the line’s position is part of the measurement.

Where in the atmosphere a line forms #

Different lines come from different heights. A weak line absorbs so little that light at its centre still emerges from deep in the photosphere, where the granules’ motions are fastest and their brightness contrast largest. A strong line absorbs so much at its centre that the light emerging there comes from hundreds of kilometres higher, where the photosphere has thinned and cooled. The same principle lets the spectrum serve as a thermometer at every height: each wavelength samples the layer where the atmosphere becomes transparent to it.

Convection does not stop abruptly at the top of the convection zone. Rising gas overshoots into the stable layers above, slowing as it goes, and above a few hundred kilometres the granular velocities have largely died away. A strong line forming up there sees much less of the convective blueshift than a weak line forming at the base. The two-stream model in the budget shows the effect: with the flows decaying over a scale height of 150 kilometres, a line formed 300 kilometres up is blueshifted by 31 metres a second rather than 231.

That was the explanation of the old discrepancies. Measurements in the 1920s by Charles St John at Mount Wilson, using many iron lines, found shifts that grew towards the predicted value for the stronger lines and fell short of it for the weaker ones; at the time the pattern was taken as evidence of something wrong with the prediction, or of pressure shifts in the solar atmosphere, or of motions nobody could see. Once granulation was understood — the first good photographs of it date from the 1950s, and the detailed theory of how it shapes lines from the 1970s and 1980s — the pattern became a calibration rather than an objection.

James Brault, in 1962, measured the strong sodium resonance lines, whose centres form high in the atmosphere, and found a shift consistent with the prediction within a few per cent. Later measurements chose lines and positions on the disc designed to minimise the convective term: the potassium resonance line at 770 nanometres, observed by resonance scattering in a laboratory cell so that the line’s own laboratory position served as the reference; the infrared oxygen triplet near the limb. Each in its way was a choice of where on the Sun, and at what height, to look.

Rotation and the face of the disc #

A second motion of the surface is far larger than the granulation and much easier to remove. The Sun rotates, once in about 25 days at the equator, and its limb moves at two kilometres a second — three times the gravitational shift.

Rotation is the easy part because it is odd: the approaching limb is blueshifted by exactly as much as the receding limb is redshifted, so the average of two points symmetric about the rotation axis removes it entirely. The convective blueshift is even. Bright gas rises everywhere on the disc, and its brightness-weighted motion is a blueshift wherever it is seen from, so averaging over the disc, or over opposite points, leaves it in place. What changes across the disc is its size. At the centre of the disc the line of sight is vertical, and the rising and sinking streams are seen head-on, at their full velocities. Towards the limb the line of sight tilts, the vertical velocities are seen at the cosine of the tilt, and at the limb itself they are seen side-on and contribute nothing.

That was the first strategy for getting rid of convection: measure at the limb, where vertical motions vanish from the line of sight and the gravitational shift, which does not care about direction, is left. The limb is a difficult place to measure. The light there grazes through the upper atmosphere along a long path and comes from higher layers, the granules are seen edge-on so that their near sides hide their far sides, and the horizontal flows inside each granule, which spread from its centre towards its edges, are now along the line of sight. These have their own shifts, and real limb measurements show a redshift exceeding the gravitational value by an amount that also depends on the line. The “limb effect”, as it was called from the 1900s onward, was one of the mysteries the gravitational redshift was confused with for decades.

Two handles on one nuisance #

The two strategies — strong lines, which form above the convection, and positions near the limb, where vertical convection is seen side-on — are independent ways of reducing the same contamination. If both converge on the same value, that is evidence that the value is gravity and not a feature of the model used to subtract the convection.

In the model the convergence is exact by construction, since the model has no horizontal flows and no corrugated surface. At disc centre the three lines disagree by more than two hundred metres a second; at the limb they agree. A measurement on the real Sun cannot expect that tidiness, but the logic is the same, and it is the logic of any measurement in which a systematic error is a function of something the experimenter can vary. If the error depends on formation height and on viewing angle, then varying both and finding a common limit is a test of whether the error has been understood. A gravitational shift that varied with formation height or viewing angle would not be gravity.

This structure, a nuisance with a known dependence on accessible variables, also appears in the stacked cluster measurement of the light that climbs out of a cluster of galaxies. There the obstacle is the galaxies’ own orbital velocities, much larger than the gravitational shift, and the handle is statistical: thousands of clusters, with motions that cancel on average and a gravitational offset that does not. For the Sun the handle is physical: the nuisance is a property of the atmosphere, and the atmosphere has layers.

Reading the answer off the strongest lines #

The most precise solar measurement takes the first strategy to its conclusion. In 2020 Jonay González Hernández and colleagues used the HARPS spectrograph, calibrated with a laser frequency comb — the instrument whose regularly spaced lines are explained in the phases that turn a glow into pulses — to measure the spectrum of sunlight reflected from the Moon. Moonlight is the Sun seen as a star, integrated over its whole disc, with the advantage that the instrument and its calibration are exactly those used for distant stars.

Hundreds of iron lines were measured, of every strength, and their shifts were plotted against their depth. Weak lines, forming low in the convection, showed small shifts; deeper lines showed larger ones; and the deepest lines, whose cores form in the upper photosphere where the granules have almost stopped, flattened towards a common value. The modelled curve above has the same shape. The published result for the gravitational redshift, from the lines whose cores form highest, was 638 ± 6 metres a second, consistent with the predicted 633.

The precision is set less by the spectrograph than by the extrapolation: how the strongest lines approach their limit depends on how convection dies out with height, and that is a property of the solar atmosphere known from three-dimensional simulations of the photosphere rather than from first principles. The measurement is a test of general relativity at the level of about one per cent, which is far weaker than clocks compared at different heights on the Earth now achieve, but it is a test in the field where the prediction was first made, at a potential three thousand times deeper than the Earth’s.

Why the Sun was the hardest case #

The gravitational redshift has been measured in four kinds of place, and the Sun is the one where the nuisance is largest relative to the signal. In the Pound–Rebka tower at Harvard the shift was two parts in $10^{15}$, but the source and absorber were nuclei bound in solids at rest in a laboratory, with no motions to speak of, and the experiment’s difficulty was only its smallness. In white dwarfs, the redshift that weighs a dead star shows shifts of tens of kilometres a second, a hundred times the Sun’s, while the convective motions of a white dwarf’s thin atmosphere are much smaller than that, so the signal dominates. In the comparison of atomic clocks at different heights the signal is again tiny and the clocks are, by design, still.

The Sun sits in the worst corner of that space: a signal of six hundred metres a second against convective motions of the same size, in an atmosphere whose structure was not understood when the measurement was first attempted. That every one of these routes gives the same answer, the potential difference times one over $c^2$, is what the clocks that must all slow together argues the equivalence principle requires: any clock, of any construction, at the bottom of a potential well must run slow by the same factor, or one kind of clock could be used to tell gravity from acceleration. Nuclear transitions in the tower, atomic lines in a white dwarf’s atmosphere and iron lines in the solar photosphere are three different clocks, and they agree.

What the two-stream model leaves out #

The figures are drawn from a two-stream model with stated parameters: a rising fraction of sixty per cent, a brightness contrast of one and a half, an upflow of one kilometre a second, a decay height of 150 kilometres and a mapping from line depth to formation height of 600 kilometres times the depth squared. Those numbers are of the right size for the solar photosphere, but they are chosen to show the mechanism, not fitted to the Sun; the real granulation has a continuous distribution of velocities and temperatures, horizontal flows that the model omits, and a corrugated surface whose near side hides its far side towards the limb. Real line formation depends on the atom’s level structure and the local temperature, not on depth alone. The model’s convergence at the limb is exact because it has no horizontal motion, and the real limb overshoots. The rotation is taken as rigid at two kilometres a second along the equator, ignoring the Sun’s differential rotation and its five-minute oscillations, which shift lines by hundreds of metres a second at any instant and average out only over hours.

Still open: how well the Sun’s atmosphere is known #

The remaining uncertainty in the solar measurement is the remaining uncertainty in the model of the solar atmosphere, and the two are now studied together. Three-dimensional radiation-hydrodynamic simulations of the photosphere reproduce the observed line bisectors and their centre-to-limb variation with remarkable fidelity, and they predict each line’s convective shift from first principles; subtracting those predictions from measured shifts gives the gravitational redshift without extrapolation. Whether the simulations are right to the few metres a second that would turn the solar measurement into a test at a tenth of a per cent depends on details of radiative transfer and magnetic fields in the simulated atmosphere that are still being refined.

The habit worth carrying away is to ask whether a systematic error has a handle. The Sun’s gravitational redshift is 633 metres a second, the same for every line, and the granulation blueshift is a few hundred, different for every line and every point on the disc — so it was found not by suppressing the convection but by varying the two things the convection depends on, formation height and viewing angle, and finding the value they both converge on. A shift that did not converge would have meant the subtraction was wrong; one that converged anywhere but 633 would have meant Einstein was.